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Thrill of Victory and Agony of Defeat:Emotional Rewards and Sales Force Compensation
Ying YangDepartment of Marketing and Entrepreneurship
University of HoustonHouston, TX 77204
Email: [email protected]: 713-743-4586
Niladri B. SyamDepartment of Marketing and Entrepreneurship
University of HoustonHouston, TX 77204
Email: [email protected]: 713-743-4568
James D. Hess*Department of Marketing and Entrepreneurship
University of HoustonHouston, TX 77204
Email: [email protected]: 713-743-4175
July 10, 2012
* Corresponding Author
Thrill of Victory and Agony of Defeat:Emotional Rewards and Sales Force Compensation
Abstract
How does the optimal design of sales contests compare with sales quota systems when
sales territories are imbalanced but you account for intrinsic agents’ emotions of pride and
disappointment in goal-attainment? First, more intensely felt pride by an underdog and
disappointment by a favorite improves the firm’s profit for either a quota or a contest. Second,
pride and disappointment have different, but specific, effects on the advantage of quota over
contest. Third, if the more personal nature of the contest that pits one person against a known
rival generates a thrill of victory and agony of defeat that is missing with quotas, contests may be
more profitable than quotas even with independent random sales. Fourth, if the strength of the
territory is the source of deep potential disappointment and the weakness of the territory is the
source of great pride, then handicapping the compensation system is unprofitable.
1
1.1 Introduction
Sales contests and bonus-quota plans are frequently used to motivate salespeople,
especially for achieving short-term sales goals (Murphy and Dacin 1998). In both the contest and
the quota systems, salespeople win a prize if they surpass a sales threshold. The compensation
systems differ in how the threshold is determined: in the contest, the threshold is the sales level
of other salespeople, and in the quota, the threshold is a predetermined sales quota.
The economic aspects of quotas and contests have a long history. The theoretical
literature on contests began with the pioneering work of Lazear and Rosen (1981), Nalebuff and
Stiglitz (1983), and Green and Stokey (1983) in economics, and was continued in marketing by
Kalra and Shi (2001). The theory of bonus-quota plans (a special case of piece-rate schemes) has
been analyzed by Kim (1997) and Oyer (2000) in economics, and in marketing by scholars like
Mantrala, Raman and Desiraju (1997), Mantrala, Sinha and Zoltners (1994) and Raju and
Srinivasan (1996) etc. Some authors, notably Lazear and Rosen (1981) and Green and Stokey
(1983) have also compared relative compensation schemes like contests with individual schemes
like piece rates.
A recent stream of research has pointed out the importance of incorporating emotions to
better understand the behavior of economic agents (Rabin 1983; Holt and Sherman 1994; Elster
1998; Loewenstein 2000). Selling is a human activity and a complete investigation of the
behavior of sales agents requires us to understand how they take into account their emotions like
pride, disappointment, envy, compassion etc., when evaluating the consequences of their actions.
The extant literature comparing a relative scheme (contest) with an individual scheme
(quota) has ignored sales agents’ emotions, and our goal in this paper is to compare these
schemes when agents’ emotions are taken into account. There is a small literature dealing with
2
emotions in contests (Kräkel 2008a; Grund and Sliwka 2005; Chen, Ham and Lim 2011).
However, making the comparison of contests with quotas requires us to also analyze a quota
scheme when agent emotions are incorporated. This is novel to the economics literature and is a
major contribution of our research. Like a contest, a quota scheme also has a natural target such
that an agent’s performance with respect to the target may give rise to emotions of pride and
disappointment. Just as in the contest the agent can feel pride or disappointment depending on
whether his sales exceeds or falls short of his rival’s, so too in a quota scheme the agent can feel
pride or disappointment if his sales exceeds the predetermined quota or not (Deci and Ryan,
1985, p.218-220; Latham, 2003).
The setting for our research is a situation where a firm is trying to design its
compensation scheme when faced with sales territories that are imbalanced. Though managers
try as best they can to balance the territories so that all have equivalent sales potential, as a
practical matter this is not always feasible and maybe too costly. Many practitioners have
commented on how unequal sales territories are in actual practice: “We have observed that sales
managers are frequently surprised to learn how unequal their sales territories are,” (Zoltners and
Lorimer 2000, p.139). There have been some attempts at incorporating territory potential in sales
response functions (Lucas et al. 1975), but the problem of properly accounting for territory
potential in designing compensation persists. Moreover, the problem of determining and aligning
territory potential is likely to endure (see Zoltners et al. 2006). Territory imbalance has the same
effect as asymmetry among agents (different experience, costs, skill levels, etc.) and agent
asymmetry is an important consideration in incentive design. Clearly any compensation scheme
must take into account territory imbalance or agent asymmetry.
3
We have two research questions. First, when territories are imbalanced, how does a
relative plan like the contest compare with an individual plan like the quota scheme when agents’
emotions of pride and disappointment are taken into account? Second, since there is imbalance
between the territories, a natural question to ask is how handicapping the contest and quota
scheme compare with the un-handicapped ones when agents have feelings of pride and
disappointment.
1.2 Overview of Results
A major finding is that when territories are imbalanced, pride in winning/making quota
increases profits from both the quota and the contest but it benefits the contest more. As a
contrast note than when emotions are ignored territory imbalance decreases profits from both
quota and the contest but it hurts the contest more (Syam, Hess and Yang 2011). In fact a major
insight we provide is when and how emotions can counter the effects of territory imbalance.
If the intensity of emotions is the same for both contests and quotas a quota system is
more profitable. However, there is ample evidence that the mano-a-mano character of a contest,
pitting one salesperson against a known, personal rival, creates more intense feelings of pride
and disappointment than the impersonal benchmark of the quota system. In that case the contest
is more profitable. This result is new to the literature comparing compensation schemes, and also
contrasts starkly with the case when agent’s emotions are ignored. The extant literature
comparing relative and individual scheme finds that when emotions are ignored, the quota
scheme is more profitable for the firm with both balanced and imbalanced territories (Lazear and
Rosen 1981; Green and Stokey 1983; Syam, Hess and Yang 2011).
The primary reason why contests may be desirable was provided by Green and Stokey
(1983), who show that relative plans (contest) can dominate individual plans (quotas) when there
4
are sizeable common shocks. We have uncovered a new rationale for contests to dominate quotas
even in the absence of common shocks. Since emotions play an important role in the behavior of
economic actors, our results can help explain the enormous popularity of contests, especially in
the sales setting.
Another salient finding in the current paper is that, when agents’ emotions are
incorporated the firm may find it better not to handicap the contest and quota scheme. When
agents’ emotions are ignored, then it is optimal to handicap the contest and quota scheme since
handicapping can overcome the deleterious effects of territory imbalance (or agent asymmetry),
and the contest and quota become as profitable as in the case with balanced territories. The
reason for our novel finding about the sub-optimality of handicapping when emotions are
accounted for is that handicapping has two contrasting effects. On the one hand, it removes the
negative effect of territory imbalance, but it also removes the impetus for agents’ emotions to
take effect. Since it is precisely the imbalance between territories that opens the door for pride
and disappointment, negating the imbalance also removes the source of these emotions. Since the
firm’s profit increases in these emotions, this is a negative effect of handicapping. We show that
when the emotions are intense enough, the firm is better off without handicapping.
Incidentally, our results provide one rationale why we see so few handicapped contests in
practice, even though they have been theoretically investigated and have been shown to increase
firm profits from a purely economic standpoint (Lazear and Rosen 1981). It has been argued that
from a practical standpoint a firm may eschew handicapping since it may not be able to handicap
properly and this would give rise to concerns of fairness, equity etc. We show that handicapping
may be undesirable even if the firm were able to handicap optimally to avoid the deleterious
effects of territory imbalance.
5
Because the contest involves interpersonal comparisons with other active participants it is
likely to generate stronger emotions than quotas, which are impersonal accounting standards. In
that case the firm will prefer not to handicap the contest even in situations where it chooses to
handicap the quota by setting territory-specific quota schemes. This may explain why
handicapped contests are much rarer than handicapped quotas.
1.3 Literature Review
As already mentioned, the extant work incorporating emotions in compensation schemes
involves contests. First, Kräkel (2008a) has investigated the emotions that come into play when
agents compare their own performance with the performance of other agents. Such interpersonal
comparisons engender feelings of pride (positive emotion) if one wins the contest, and feelings
of disappointment (negative emotion) if one loses. With heterogeneous agents who differ in their
innate abilities, the author finds that given exogenously defined prizes the agents’ efforts are
increasing in both pride and disappointment. This result rationalizes the empirical fact that agents
are found to overexert themselves compared to the predictions of standard tournament theory. He
has also endogenized the prize structure. Kräkel (2008b) carried out a similar analysis of salary
and commission. Though known asymmetry among agents could induce the firm to offer
handicapped contests as an optimal mechanism (a la Lazear and Rosen 1981), unlike us Kräkel
does not investigate handicapping.
Second, Grund and Sliwka (2005) investigate envy and compassion in contests.
Compassion is felt by the winner of the contest and envy is felt by the loser. Both emotions come
into play because of the agents’ aversion to inequity, which is felt because a contest always has a
winner and a loser. Indeed, the amount of inequity is operationalized via the difference between
6
the winning and losing prizes. They find that, in the partial equilibrium where prizes are
exogenously defined, the optimal efforts are higher in a contest with inequity-averse agents
compared to a standard contest where agents only have self-regarding preferences. However, in
the full equilibrium with endogenous prizes chosen by the principal, the effort in a contest with
inequity-averse agents is smaller compared to a standard contests. Moreover, the principal’s
profit is smaller in tournaments with inequity-averse agents than in tournaments where agents
have self-regarding preferences.
Third, Gill and Stone (2010) study tournaments where agents like to receive ‘just deserts’
for their efforts. If an agent exerts more effort than his rival, then he feels that he deserves more,
and an agent’s utility is the sum of his monetary payoff and a comparison of this payoff with an
endogenous reference point that depends on both agents’ efforts. The authors’ main goal is to
demonstrate that when agents care about receiving just deserts, then a tournament with
symmetric agents can give rise to asymmetric efforts in equilibrium, contrary to standard
tournament theory.
Fourth, Chen, Ham and Lim (2011) experimentally investigate agent emotions in a multi-
person tournament with asymmetric agents. Like Kräkel (2008a) they too find that incorporating
emotions in tournaments allows researchers to rationalize the over-exertion of effort in the
laboratory compared to the prediction of standard tournament theory. They also investigate how
changing the number of prizes in a multi-person tournament changes the efforts exerted by the
agents. However, they do not derive nor use optimal prizes in their experiments and their results
are derived using exogenously given prizes.
We assume some salespeople have advantages owing to working in the better territory.
Thus there are feelings of pride when the agent in the weaker territory wins the contest and
7
feelings of disappointment when the agent in the stronger territory loses the contest. Similarly,
there are feelings of pride when the agent in the weaker territory makes quota and feelings of
disappointment when the agent in the stronger territory fails to make quota.
Our contributions to the literature are, first, to analyze the effect of emotions on both
quota and contest and to contrast the resulting differences in the profits of the two compensation
systems (see Table 1). Second, we also contribute by investigating the effects of handicapping
the contest and quota when agents are asymmetric and their emotions are taken into account.
Homogeneous Sales Response Heterogeneous Sales ResponseOptimal Payment
Based Upon:Only
Financial RewardsEmotional and
Financial RewardsOnly
Financial RewardsEmotional and
Financial Rewards
Contest
Lazear & Rosen (1981)Green & Stokey (1983)Nalebuff & Stiglitz (1983)Kalra and Shi (2001)Syam, Hess, & Yang (2011)
Grund & Sliwka (2005)Gill & Stone (2010)
This paper
Lazear & Rosen (1981)Meyer (1992)Syam, Hess, & Yang (2011)
Kräkel (2008a)Gill & Stone (2010)
This paper
Handicapped Contest Syam, Hess, & Yang (2011)
Thispaper
Syam, Hess, & Yang (2011) Thispaper
Quota or Commission
Lazear & Rosen (1981)Green & Stokey (1983)Nalebuff & Stiglitz (1983)Kim (1997)Syam, Hess, & Yang (2011)
Kräkel (2008b)This paper
Syam, Hess, & Yang (2011) Thispaper
Handicapped Quota Syam, Hess, & Yang (2011)
Thispaper
Thispaper
Comparisons
Contest vs. Quota Syam, Hess, & Yang (2011)This
paper Syam, Hess, & Yang (2011)This
paper
Contest vs. Commission
Lazear & Rosen (1981)Green & Stokey (1983)Nalebuff & Stiglitz (1983)Syam, Hess, & Yang (2011)
Syam, Hess, & Yang (2011)
Table 1. Models of Multi-Agent Payment Systems:Asymmetric Sales Reponses and Emotional Rewards
2. Sales Quota and Sales Contest
Suppose a risk-neutral firm employs two risk-neutral salespersons who will exert efforts to
sell the firm’s goods. Each salesperson is assigned to a separate territory, i=1, 2. The firm cannot
observe the levels of salespersons’ effort, ei, but can observe dollar sales, si. The sales in a
8
territory depend not only on the salesperson’s level of effort but also on the territory potential.
Without loss of generality, we assume territory 1 is the stronger territory. In particular, the sales
in territory 1 is s1=k+e1+ε1 , and the sales in territory 2 is s2=−k+e2+ε2 , where 0<k<½ is
the advantage of territory 1 over territory 2, ei is the effort level of salesperson in territory i, and
a random component of sales, i, is uniformly distributed on the interval [- ½ , ½ ]. Sales in
territory 1 are shifted up by k and the sales in territory 2 are shifted down by the same amount;
we limit the imbalance k to ½ so that sales are never negative with certainty. The functional
forms of the sales response functions are exactly as in Kräkel (2008a) who interprets the
exogenous constant as agents’ ability, whereas we interpret it as territory imbalance. The reader
could think of an average level of sales across the two territories with one territory enjoying an
advantage over the other.
The random variable, i, independent and identically distributed across different territories,
reflects the sales influenced by territory-specific environmental shocks outside the salespersons’
control. The assumption of independence is chosen to level the playing field to focus just on
territory imbalance. If there were common shocks, they would cancel in a comparison of sales in
a contest but would not disappear in a quota system.
2.1 Bonus-Quota
Consider a quota payment system.1 In a quota system the salesperson is entitled to a
bonus only if sales exceeds a prespecified quota, Q, and not otherwise:
Pay (si;Q , B)=¿ {0 if si<Q ,¿ ¿¿¿ (
1 In this section we assume that the principal sets the same bonus and quota for both territories. Thus both the contest and the quota have the same number of decision variables: S and B for the contest and B and Q for the quota. A situation wherein the principal sets territory specific quotas is called ‘handicapped’ quotas just as a contest which accounts for territory imbalance is called a ‘handicapped’ contest. A major result of our paper is to point out the sub-optimality of handicapping quotas when agents’ emotions are taken into account. This is done in Section 4.
9
The quota system has two facets to control, B and Q, the same number as in the sales contest; for
analytic convenience we push baseline salary into the background.
We now incorporate agents’ emotions in the analysis. The salesperson derives benefits
from the pay received for working, but also has intrinsic utility from a job well-done.
Specifically, if the salesperson is able to achieve the bonus there is pride, a positive emotion felt
when a sales goal is accomplished. On the other hand, if the salesperson fails to earn a sales
bonus there is disappointment, a negative emotion felt when there is failure to achieve a goal.
This operationalization of emotions is the same as in Kräkel (2008a) who investigated only
emotions in contests whereas we analyze emotions in both a contest and a quota scheme.
Throughout this paper we will denote the utility associated with pride by the symbol and the
utility associated with disappointment as . The salesperson in the stronger territory 1 may have
different emotions than the salesperson in the weaker territory 2, so we will include a subscript in
pride and disappointment to indicate the territory. Finally, it is possible that the emotions depend
on whether the goal is to achieve a firm given quota or to outsell a fellow salesperson, so we will
add a superscript to indicated whether the goal is quota or sales contest.
The combination of extrinsic financial reward and intrinsic emotional reward leads to
utility in a quota:
U i=¿ {0 - δ iQ , if si<Q ¿ ¿¿¿ (
Efforts are costly to the salespeople and rise quadratically with effort in our model, ½ ei2.
Achieving quota is never certain. The probability that the salesperson assigned the
stronger territory 1 can make the sales quota is P1=½ + k - Q + e1, while the probability in
territory 2 is smaller, P2=½ - k - Q + e2. The expected net utility of the salesperson assigned to
10
territory i is EUi=(B+iQ)Pi+(0-i
Q)(1-Pi)– ½ei2. Notice that in the expected utility the agent fully
anticipates the emotions that will be felt in the future depending upon success or failure in the
achieving quota. Such affective forecasts have been studied in other contexts (Gilbert and Wilson
2000, Patrick, Chun, MacInnis 2009).
The efforts that maximize these expected net utilities are
e1=B+π1Q+δ1
Q and e2=B+π2
Q+δ2Q
. (
Notice that while these partial equilibrium efforts increase with emotional rewards (obtaining
pride or avoiding disappointment), they do not depended on the quota or degree that the
territories are imbalanced. This is because added effort moves sales the same amount whether
the current sales are near or far from the quota.
With the efforts of (, the probability of making quota is higher in the strong territory 1 if
and only if
{ k+π1Q+δ1
Q }−{−k+π2Q+δ 2
Q }≥0 . (
Recall that territory 1 has a sales advantage of +k and territory 2 has a sales disadvantage –k.
Emotions associated with the job tend to increase the advantage in the strong territory and reduce
the disadvantage in the weak territory. That is, intrinsic emotions in the strong territory
implicitly increase imbalance while emotions in the weak territory implicitly reduce imbalance.
Inequality ( states that intrinsic emotional rewards are not so intense as to reverse the basic
territorial imbalance.
Chen et al. (2011) say, “For favorites, we assume that they expect themselves
11
and other favorites to win. Hence, receiving a prize does not provide favorites any additional
psychological utility gains… Underdogs, on the other hand, are assumed to expect to lose
because they compete with an initial disadvantage. Hence, losing in the tournament
does not bring any psychological disutility.” (p.872). Kräkel (2008a) agrees with this when he
says, “If, for example, agents do not feel strong emotions when losing (winning) against more
(less)able opponents as this outcome is the most likely one, then the only emotions that matter
arise when strong contestants lose against less able ones and when low able agents win against
predominant opponents.” (p. 209). As a consequence, we will reduce the mathematical notation
from this point forward by assuming that in the stronger territory 1 the salesperson feels no pride
from making quota and that in the weaker territory 2 the salesperson feels no disappointment:
thus, 1Q =2
Q =0.
The maximum utility in the weaker territory 2 is
EU 2=(B+π 2Q)( 1
2−k−Q)+ 12 (B+π2
Q )2 . (
Set this to zero, solve this for the quota that motivates participation in the weaker territory,
Q= 12−k+ 1
2 (B+π2Q ). Because it is more difficult to motivate participation in the weaker
territory, the quota depends upon pride, specific to the weaker territory. Greater emotions in the
other, stronger territory have no direct effect on quota. The probability that both salespeople
make quota is P1+P2=( 1
2+k−Q+B+δ 1Q )+( 1
2−k−Q+B+π2Q ) , which after substituting the
quota reduces to P1+P2=B+δ 1Q+2 k . Because the quota applies to both territories, an
increase pride 2Q has no net effect on the probability that both territories make quota.
We can write profit only as a function of the bonus:
12
EΠ Q=e1+e2−B (Pr (1 makes Quota)+Pr (2 makes Quota))=2B+δ1
Q+π2Q−B (B+δ1
Q+2 k ) .(
The optimal bonus is
B∗¿1−k− 12 δ1
Q . (
The bonus is reduced if there is greater disappointment in the strong territory because this leads
to greater effort in that territory and thus more frequent payouts of the bonus. On the other hand
greater pride in the weaker territory is neutralized by an increase in quota, so the bonus is
independent of 2C.
Back substitution gives profits, efforts and quotas in the full equilibrium:
EΠQ∗¿=δ 1
Q+ π2Q+(1−k−
δ1Q
2 )2
¿(
e1∗¿1−k+ 12 δ1
Q , (
e2∗¿1−k+π2Q−1
2 δ1Q , (
Q∗¿ 32 (
23−k )− 1
4 δ1Q+ 1
2 π2Q . (
The envelope theorem says that emotions have a direct effect on profits,
∂ EΠ Q
∂Emotion , and
an indirect effect,
∂EΠ Q
∂B∂B
∂Emotion , but this indirect effect vanishes because the bonus is
optimally chosen to have zero marginal effect. As a result it is easy to see from ( that the direct
effect of pride is 1 and the direct effect of disappointment is 1-B, both of which must be positive.
This follows because effort rises directly with emotions in ( and the probability of paying the
bonus rises with disappointment. Finally, suppose that overall emotional intensity rises in both
territories so that π2Q=π20
Q +μ and δ 1Q=δ10
Q +μ . By the envelope theorem, the direct effect of
13
increases in emotional intensity is 2-B and this will be greater than 1. Proofs are found in the
Appendix.
Proposition 1: In the full equilibrium, after accounting for optimization of the bonus and quota by the firm, the profits of a quota system rise with a ceteris paribus increase in either pride in the weaker territory or disappointment in the stronger territory and if emotional intensity increases an equal amount in both territories, profits rise by a multiple that exceeds 1.
2.2 Sales Contest
Now consider a sales contest between two salespersons with a bonus paid to the one with
the higher sales:
Pay (si; s j ; S , B )=S+¿ {0 if si<s j ,¿ ¿¿¿ (
The contest has two prizes, S and S+B, with S given to the contest loser and S+B given to the
contest winner.
The combination of extrinsic reward and intrinsic emotional reward leads to utility in a
contest of
U i=¿ {S - δ iC , if si<s j , ¿¿¿¿ (
Winning the sales contest is not certain because of the random components of sales. The
probability that the salesperson in territory 1 wins the sales contest is Prob(s1≥s2) = P1= 1-
½(1+e2-e1-2k)2, given e1-(1-2k) ≤e2≤e1+2k. As a result the expected net utility of the salesperson
assigned to territory 1 is
EU1=S-1C+(B+1
C+1C)P1- ½ e1
2. (
14
The expected net utility in territory 2 is
EU2=S-2+(B+2C+2
C)(1-P1)- ½ e22. (
Nash equilibrium efforts by the two rival salespeople are found by simultaneously setting
the derivatives of the expected utility with respect to the appropriate effort equal to zero and
solving the two equations to get
[e1
e2 ]= 2(1/2−k )2(1/2−k )+{k+π 1
C+δ1C }−{−k+π2
C+δ2C }[B+π1
C+δ1C
B+π 2C+δ2
C ] (
Notice two things in this formula. First, both extrinsic and intrinsic emotional rewards
drive effort to win the contest. Second, effort is also dependent on total imbalance of the
territories correcting for intrinsic emotions. Specifically, the term {k+1C+1
C} - {-k+2C+2
C} is
a measure of the emotion-adjusted territorial imbalance, as opposed to k – [-k] = 2k, the ‘real’
territorial imbalance. One can show that both efforts decrease as territories become more
imbalanced. The salesperson in the weaker territory is discouraged by greater imbalance; the
salesperson in the stronger territory mimics the other’s effort and depends on the sales advantage
to win the contest.
The full equilibrium efforts must take into account the optimal bonus, B, which remains
to be solved for. As we did in the analysis of the quota, we will now assume that in the stronger
territory the salesperson feels no pride from winning a contest because it is the favorite and that
in the weaker territory the salesperson feels no disappointment in losing because it is the
underdog.
The resulting probability that territory 1 wins the contest is
15
P1=1−2( 1/2−k1+δ 1
C−π2C )
2
.(
This is independent of the bonus because extrinsic rewards drive both salespeople equivalently.
Given this and the efforts (, the expected utility of the salesperson in the weaker territory 2 is
EU 2=S+2( 1 /2−k1+δ1
C−π2C )
2
(B+π 2C−(B+π 2
C )2) . (
The participation of this salesperson requires that this be at least zero. Solving for the salary that
is just sufficient for participation gives:
S=−12 ( 2(1/2−k )
1+δ1C−π2
C )2
(B+π 2C−(B+π2
C )2) . (
The expected profit consists to the total expected efforts (terms k and –k cancel and the
expected value of i is zero) minus two salaries and one bonus (only one salesperson wins the
contest). This permits us to write profit as a function of the bonus for winning the contest:
EΠ C=e1+e2−2 S−B
¿2(1/2−k )1+δ 1
C−π2C (2 B+π 2
C+δ1C ) +(2(1/2−k )
1+δ 1C−π2
C )2
(B+π2C−(B+π2
C )2 )−B . (
The expected profit maximizing bonus is therefore
B∗¿1−π 2C−1
2 ( δ1C−π2
C+2 k2(1/2−k ) )
2
. (
If disappointment in the strong territory is more intense, this implicitly increases
imbalance and, as seen above, discourages effort. The firm has less incentive to pay a bonus in
this less effective contest. If pride in the weaker territory is more intense, there are two effects: a
direct reward effect and a territory balancing effect. Because the intrinsic reward (pride) is
16
greater in territory 2, the firm can reduce its extrinsic reward B without fear of losing the
employee. On the other hand, more intense emotions in the weaker territory implicitly makes the
territories more balanced and as discussed above, this tends to encourage effort and hence give
the firm an incentive to pay a larger bonus. The net effect on the bonus is indeterminate.
Back substitution from the optimal bonus in ( gives the full equibrium efforts:
e1∗¿1+2( 12−k )−
12−k
1+δ1C−π2
C−14
1+δ1C−π 2
C
12−k ,
(
e2∗¿1+12−k
1+δ 1C−π2
C−14
1+δ1C−π2
C
12−k ,
(
Substituting the efforts, the optimal maximized profits with a contest equals,
EΠ C∗¿ π2C+2( 1
2−k )+ 1
4 ( 2( 12−k )
1+δ1C−π2
C+2−
1+δ1C−π2
C
2( 12−k ) )
2
−2 ( 1
2−k )1+δ1
C−π2C
. (
Consider the equilibrium efforts in ( and (. Notice that they depend upon the difference
between the disappointment in the strong territory and the pride in the weak territory. Hence,
once the firm has adjusted the financial compensation, a uniform increase in the emotional
intensity of salespeople will have no effect on their effort choices. What about independent
changes of the emotions in the two territories? If disappointment in the strong territory increases,
the motivation to avoid the loss of the contest tends to increase effort in territory 1 but more
imbalanced territories tend to reduce effort (recall that emotions in the strong territory increase
effective imbalance). One can show that so long as the implicit imbalance remains (see
inequality (), the latter dominates and effort decreases in territory 1. It is also clear that the
increased implicit imbalance discourages effort in the weaker territory 2. Thus, greater
disappointment in the strong territory decreases effort in both territories. If pride in the weak
17
territory increases, then effort goes up in that territory as the salesperson seeks the more intense
emotional rewards in a more implicitly balanced contest (emotions in the weak territory reduce
effective imbalance). The salesperson in the stronger territory must mark his rival, counting on
territorial imbalance to win the contest. That is, greater pride in the weak territory stokes effort in
both territories.
Turning to profits, notice that the first term in the profit formula ( is 2C. This appears
because the firm needs to convince the salesperson in the weak territory 2 to participate. If this
salesperson has intrinsic desires to win the contest (that is, 2C), the firm need not provide as
large a financial reward to maintain participation and this cost reduction increases profits. We
can show that an increase in pride in territory 2 increases profits in the contest. Disappointment
in the strong territory only indirectly influences profits by implicitly increasing the imbalance of
the territories: k+1C –{-k+2
C} increases with disappointment. In a model with imbalance but
no emotions, Syam, Hess and Yang (2011) show that profits in a contest fall with imbalance
because this discourages sales effort. This carries over for disappointment in the strong territory
and an increase in 1C reduces profits when k is not too large. If both disappointment and pride
intensify equally, this has no effect on implicit territory imbalance, but the firm can reduce the
bonus without losing the salesperson in the weak territory; this improves the firm’s profits. Thus
we have the following.
Proposition 2: In the full equilibrium, after accounting for optimization of the compensation by the firm, the profits of a contest increases in pride in the weaker territory and decreases in disappointment in the stronger territory (if imbalance is not too large). If emotional intensity increases by an equal amount in both territories, profits rise by a multiple of 1.
18
It is important to point out two significant differences between our results and those in
the extant literature which incorporates emotions in contests. First, Chen, Ham and Lim (2011)
find that the agents’ efforts depend on emotions. However, they have only focused on the agent’s
efforts given exogenous prizes and have not endogenized the prizes offered by the principal. In
other words, they have looked at the equivalent of the partial equilibrium equation ( rather than
full equilibrium equations (-(. In the full equilibrium, after accounting for optimization of the
prizes by the firm, the efforts of both salespeople increase in the pride in the weaker territory and
decrease in the disappointment in the stronger territory. However, an equal change in emotional
intensity in both territories has no effect on the efforts expended.
Second, Kräkel (2008a) has endogenized tournament prizes, but all he says is that pride
and disappointment may or may not improve the firm’s profits (Proposition 2, p.210). Essentially
his result says that anything can happen and so is very non-specific. Improving on Kräkel’s
result, we can provide sharper predictions regarding the effect of emotions on profits from the
contest: increase in pride in territory 2 increases profits in the contest whereas increase in
disappointment in territory 1 decreases profits.
3. Profits of Contest versus Quota with Pride and Disappointment
The expected profits for quota and contest have been expressed in equations ( and (,
respectively, as functions of territory imbalance, k, disappointment in the strong territory and
pride in the weak territory. To simplify exposition we will begin the comparison by assuming
that pride of winning the contest is equally intense as pride in making quota, π2Q=π2
C≡π2 and
similarly for disappointment, δ 1Q=δ1
C=δ1 . It is very plausible that emotions are more intense in
a contest, where a salesperson is trying to outsell someone that they know personally, than in a
19
quota system where a salesperson is just trying to achieve a sales target set by the firm. Later we
will address the more reasonable situation where pride and disappointment are larger in a contest
than in a quota.
Not all values of the parameters k, 2, 1 are practical. We would like to ensure that the
probabilities of making quota are positive fractions. Specifically, for the quota system we would
like emotional intensity to be small enough that the strong territory is not one hundred percent
certain to make quota: P1=12+
32 k+ 3
4 δ 1−12 π2≤1 . Also, all efforts must be positive. These
restrictions on the parameters essentially ensure that the interior solutions are valid and they
imply that neither emotions nor imbalance should be too large, as seen in Figure 1.
Figure 1: Limits on Emotional Intensity
Let us begin by assuming there are no intrinsic emotional motivations. The simplified
profits of quota and contest are EΠ Q∗¿=( 1−k )2 ¿ and
EΠ C∗¿(1−k− k2( 1
2−k ))2
. Clearly,
20
A
2
0
Pride, 2
Disappointment, 12/3-2k 2(1-k)
2k
when there is no thrill of victory or agony of defeat with imbalanced territories, the quota is more
profitable than the contest. This was previously reported in Syam, Hess and Yang (2011). Their
basic logic was that with imbalanced territories, the salesperson in the stronger territory cannot
see how he could lose and the salesperson in the weaker territory cannot imagine how it can win
a sales contest. All the salesperson in the stronger territory needs to do is to mimic the effort
exerted by the agent in the weaker territory and simply allow the territorial advantage to win the
contest. On the other hand, such imitation plays no role if the bonus is rewarded for
independently achieving a predetermined quota.
What happens to these profits if the emotions associated with winning or avoiding losing
are more intense? As seen in Propositions 1 and 2, if the salesperson in the strong territory is
more intensely disappointed in not achieving their goal, the profit of the quota goes up and the
profit of the contest goes down. The basic intuition for this is that increasing disappointment in
the strong territory implicitly makes the two territories more imbalanced and this has a greater
negative impact when one territory’s sales are contrasted to the other (contest), rather than to a
predetermined standard (quota). Because the quota system begins with an imbalance-induced
advantage, an increase in intensity of disappointment makes the advantage even greater.
One the other hand, if the salesperson in the weak territory is more intensely proud profits
rise in both the quota and the contest. In the contest there are two complementary driving forces:
increased intrinsic rewards and improved balance of the territories (recall that pride reduces
implicit imbalance). These two positive forces amplify each other and so increased pride has a
larger effect in a contest compared to a quota. As a result, as 2 increases, the advantage of the
quota due to territory imbalance shrinks. However, for all practical levels of pride, the quota’s
imbalance-generated advantage is not completely overcome by emotional rewards.
21
Finally, what happens to quota’s advantage if pride and disappointment intensify in equal
amounts? This gives a differential benefit to the quota, enlarging its imbalance-based advantage.
The reason is that an equal increase in 2 and 1 has no effect on the implicit territorial
imbalance, k+1-{-k+2}. This reduces the impact of pride in the contest and thus the gap
between contest and quota widens.
Proposition 3: If emotional levels are independent of the payment system and territories are imbalanced then,a. The profit advantage of the quota over the contest decreases with a ceteris paribus
increase in pride in the weaker territoryb. The profit advantage of the quota over the contest increases with a ceteris paribus
increase in disappointment in the stronger territory (if imbalance is not too large). c. If emotional intensity increases an equal amount in both territories, the gap between
quota and contest profits widens.
The result in Proposition 3 is new since the extant literature comparing different schemes
has ignored emotions. Part (a) says that pride in the weaker territory increases profits for both the
contest and the quota but it benefits the contest more. In contrast Syam, Hess and Yang (2011)
find that when emotions are ignored, territory imbalance decreases profits for the contest and the
quota but it hurts the contest more.
One might expect that emotions are more intense in a contest than quota, because the
contest can get personal: “I really want to win the contest to show Tom who is the better
salesperson!” Can the advantage of the quota due to imbalanced territories be overcome by these
intrinsic motivations? The answer is, “Yes.” Suppose we begin without emotions at point 0 in
Figure 1, and allow pride and disappointment to intensify only in the contest.2 That is, suppose
that emotions for the contest enlarge along the dashed 45o line until they reach the level
corresponding to point A. At this point, the emotions are δ1C=π2
C=2(1−3 k ) . It is 2 The emotions in the quota are set to zero only for analytical simplicity. Clearly our result will hold as long as the contest engenders sufficiently stronger emotions compared to the quota.
22
straightforward to show that the profits of the contest at this point A exceed the profits of the
quota without emotions, so long as imbalance is not too large (k≤0.2).
Proposition 4: Suppose that the quota system does not engender pride and disappointment, but that the mano-a-mano character of the contest creates feelings of pride and disappointment. Even though the quota system has a natural advantage with imbalanced territories, the emotional motivations of the contest can make it the more profitable compensation system.
This result is also new to the literature. In the absence of emotions, the extant literature
comparing relative and absolute schemes finds that the quota is either equivalent to the contest or
dominates it (Lazear and Rosen 1981; Green and Stokey 1983; Syam, Hess and Yang 2011),
unless there are sizeable common shocks. We have shown that when agents’ emotions are
properly accounted for in compensation plans, the contest can dominate the quota even in the
absence of common shocks. Of course, if there are common shocks this advantage of the contest
will be even more pronounced.
4. Handicapping the Territories
Suppose that the territories can be handicapped so that the imbalance is contractually
eliminated. Handicapping of the contest can be done if the firm compares the sales from the two
territories, explicitly accounting for the fact that territory 2 is disadvantaged, a la Lazear and
Rosen (1981). Handicapping of the quota can be done simply by setting territory-specific quotas.
As shown in the Appendix, optimally handicapping the contest and quota neutralizes the effect
of imbalance, and the agents’ efforts and firm’s profits are those that would obtain if territories
were balanced.
However, this may have an unintended consequence of eliminating the strong emotions
associated with the job, since it is the imbalanced nature of the territories that engenders the
23
strong emotions as alluded to in Section 4. Imbalance in territories creates an underdog versus
favorite situation and gives rise to the strong emotions associated with winning or losing (see
also Kräkel 2008a). Without imbalance there can still be emotions but at the minimal level which
can be normalized to zero. Suppose that the firm can set k=0 at the consequence of also setting a
common emotional intensity δ1C=π2
C=μC equal to zero. The profits in a contest would change
from μC+(1−k−2/ (1−2 k ))2 to 1, a drop in profits if the pride were large. This is true if
μC≥1−(1−k− k1−2 k )
2. (
As seen in Figure 2a, for large emotional intensity, it does not pay to handicap a sales contest.
Figure 2: Region of Unprofitable Handicapping of Contest and Quota as a Function of Emotional Intensity and Territory Imbalance
24
b.a.
TerritoryImbalance, k
Unprofitable to Handicap Quota
0.20
0.8
Emotional Intensity, Q
Unprofitable to Handicap Contest
0.20
0.8
Emotional Intensity, C
TerritoryImbalance, k
The profits in the quota system would change from 2μQ+(1−k−μQ
2 )2
to 1 if the
territories were balanced by a handicapped quota system. This would be a drop in profits if the
emotions were large:
μQ≥2√1+4 k−2 (1+k ) . (
As seen in Figure 2b, for large emotional intensity, it does not pay to handicap quotas.
We have assumed in this section until now that the emotions in a contest are just as
intense as those in a quota. Contrary to the classic line from the movies, “It’s not personal; it’s
strictly business,” a sales contest can be very personal. Beating a notable colleague in a sales
contest is likely to give rise to stronger emotions than merely exceeding an impersonal sales
quota. If so, it is possible that the firm would want to handicap a quota system, but not a contest.
Proposition 5: If emotional intensity is large but would vanish if territories are balanced, handicapping the territories is unprofitable for either contest or quota. If, in addition, the mano-a-mano character of the contest makes the intensity of pride and disappointment much greater in contests than with quotas, the firm profits from handicapping a quota but not from handicapping a contest.
Thus, by incorporating agents’ emotions into contests and quotas, our theory provides a
rationale for each of two commonly observed anecdotal practices regarding handicapping of
compensation systems: first, we often do not see handicapped contests and second, it is much
more common to see handicapped quotas rather than handicapped contests. Of course, if the
emotional intensity is small, the firm will still find it optimal to handicap the territories.
25
5. Conclusions
In this research we introduced the emotions of pride and disappointment in the
comparison of sales contests and quotas when sales territories are imbalance. Specifically, since
an agent in the stronger territory was expected to win, she feels disappointment if she loses the
contest or fails to make quota. Similarly, an agent in the weaker territory feels pride if she wins
the contest or makes quota since she was expected to lose. Though there has been some work in
incorporating emotions in contests, the incorporating of emotions in quotas is new to the
literature.
One major result is that when emotions in a contest are intense enough the contest can be
more profitable than the quota. This result is novel since extant work finds that, in the absence of
common shocks, the quota dominates the contest on purely economic grounds when agents’
emotions are not accounted for. Thus, we provide a new rationale for the popularity of contests
even when there are no common shocks in the environment.
Another result is that when emotions are intense enough, the firm will not want to
handicap the contest and quota system. This result too runs counter to the literature which finds
that handicapping is always beneficial on economic grounds since it can overcome the negative
effects of territory imbalance which always reduces agents’ efforts. The reason is that territory
imbalance paves the way for agents’ emotions to take effect, and by handicapping the
compensation schemes to negate the effect of imbalance the firm also marginalizes the role of
emotions. Since profits increase in emotions, the result follows. If the emotions for contests are
more intense than for quotas, as they could be due to their interpersonal nature, then the firm will
handicap a quota even when it chooses not to handicap the contest. This provides a rationale for
why we see many handicapped quotas but very few handicapped contests.
26
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28
Appendix
Quota
Achieving quota is never certain. The probability that the salesperson assigned the stronger
territory 1 can make the sales quota is P1=½ + k - Q + e1, while the probability in territory 2 is
smaller, P2=½ - k - Q + e2. The expected net utility of the salesperson assigned to territory i is
EUi=(B+i)Pi+(0-i)(1-Pi)– ½ei2. Specifically, EU1=(B)P1+(0-1)(1-P1)– ½e1
2 =-1+(B+1)(½ + k
- Q + e1) – ½ei2. Take the derivative and set to zero,
∂EU 1
∂e1=B+δ1−e1=0
, to get the optimal
effort e1=B+1. Similarly for territory 2, EU2=(B+2)P2+0(1-P2)– ½e22=(B+2)( ½ - k - Q + e2)–
½e22 and the first order condition is
∂EU 2
∂ e2=B+π2−e2=0
and the resulting effort is e2=B+2.
Back substituting in territory 2 give EU 2=(B+π 2)(12−k−Q )+ 1
2 (B+π2)2=0 .
Solve for Q: Q=12−k+ 1
2 (B+π2 ) . Substituting Q gives the expected profit is
EΠ Q=e1+e2−B Pr (1 makes Quota)−B Pr (2 makes Quota )=2 B+δ 1+π2−B (1
2 +k−Q+B+δ 1+12−k−Q+B+π 2)
=2 B+δ 1+π2−B (1−2 Q+2B+δ1+π2 )
=δ1+π 2−B(−2+1−2( 12−k+ 1
2 (B+π2 ))+2B+δ1+π2 )
=δ1+π 2−B(−1−2( 12−k )+B+δ 1)
=δ 1+π2+B (2(1−k )−B−δ 1) .
The first order condition
∂EΠ Q
∂B=2(1−k )−2 B−δ1=0
, yields the bonus B=1-k -½ , and
back substituting gives
EΠ Q=δ1+π2+(1−k−δ1 /2 )(2(1−k )−(1−k )+δ1 /2−δ1)=δ 1+π2+(1−k−δ 1/2)
2 ,
29
e1∗¿1−k+ 12 δ1>0 ,
e2∗¿1−k+π2−12 δ1>0 ,
Q∗¿1−32 k− 1
4 δ 1+12 π2>0 ,
0<P1=12+
32 k+ 3
4 δ1−12 π2<1 ,
0<P2=12−
12 k− 1
4 δ1+12 π2<1 .
Note that we have to restrict k, 1, and 2 so that efforts, bonus, quota, probability of making
quota are all sensible. First, we need P1<1 or π2>−1+3 k+ 32 δ 1 . We also need P2<1 or
π2<1+k+ 12 δ 1 . Finally we need π2<2 k+δ1 for territory 1 to be stronger. Note that the
first two inequalities cross at 1=2(1-k). But the first and third inequalities also cross at this
point. Since the third inequality is stricter than the second, we can ignore the second.
Contest
Expected utilities are of the form EU2=S+(B+2)(1-P1)- ½ e22 while probability of winning the
contest are Prob(s1≥s2) = P1= 1-½(1+e2-e1-2k)2. The utility maximizing efforts of both
salespeople are [e1
e2 ]=2(1/2−k )1+δ1−π2 [B+δ1
B+π2 ] .Back substituting gives
EU 2=S+(B+π2 )12 (1−2 k+e 2−e1 )2−1
2 e22
¿ S+2(B+π2 )(12−k+
12−k1+δ1−π2
(π 2−δ1 ))2−1
2e
22
¿ S+2(12−k1+δ1−π2
)2
[(B+π2 )− (B+π 2)2 ]
30
Solve this for salary and substitute into the expected profits
EΠ C=e1+e2−2 S−B=212−k1+δ 1−π2
(2B+π2+δ1 )
+4 (12−k1+δ 1−π 2
)2
(B+π2−(B+π2 )2)−B
The first order condition for the bonus is
∂EΠ C
∂B=4
12−k
1+δ 1−π2+4(
12−k
1+δ1−π2)2
(1−2(B+π 2))−1=0
1+δ1−π212−k
+1−2(B+π2 )−14 (1+δ1−π2
12−k )
2
=0
21+δ1−π2
1−2 k+1−(1+δ1−π2
1−2 k )2
=2(B+π 2)
¿−1+21+δ1−π 2
1−2 k−(1+δ1−π 2
1−2 k )2
+2=2(B+π2 )
Hence the optimal bonus is
B∗¿1−π 2−12 (1+δ1−π2
1−2 k−1)
2
=1−π2−18 ( δ1−π 2+2 k
1/2−k )2
.
Maximized profit is
EΠ C=212−k1+δ1−π2
(2(1−18 (δ1−π2+2 k
1/2−k )2
)+δ1−π 2) +
4 (12−k1+δ1−π 2
)2
(1−18 (δ 1−π2+2k
1/2−k )2
−(1−18 (δ1−π2+2k
1/2−k )2
)2)−1+1
8 (δ1−π2+2k1 /2−k )
2
+π2 ,
31
which can be simplified to
EΠ C=2( 12−k )−
2( 12−k )
1+δ1−π2+ 1
4 ( 2 ( 12−k )
1+δ1−π2+2−
1+δ1−π2
2( 12−k ) )
2
+π 2 ,
e1∗¿1+2( 12−k )−
12−k
1+δ1−π2−
14
1+δ1−π 212−k ,
e2∗¿1+12−k
1+δ1−π 2−
14
1+δ1−π212−k .
Proof of Proposition 1.
EΠ Q=δ1+π2+(1−k−δ1 /2 )2 . Taking derivatives gives
∂EΠ Q
∂ π2=1>0
and
∂EΠ Q
∂ δ1=1−δ1(1−k−δ1 /2)=1−δ 1(1−k )+δ
12 /2=(δ12−2(1−k )δ1+2)/2=((δ1−1+k )2+2−(1−k )2 )/2 .
Since 0≤k≤1/2, 1-k<1, so this derivative with respect to δ 1 is positive.
Suppose that π2=π
2o+μ , and δ1=δ1o+μ
. Then
∂EΠQ
∂ μ=2−δ1 (1−k−δ1 /2)=2−δ1(1−k )+δ
12 /2=1+(δ12−2(1−k )δ1+2)/2=1+((δ1−1+k )2+2−(1−k )2) /2 >1 .
Proof of Proposition 2.
Proof that EC increases with 2 : The expected profit is
EΠ C=π2+2( 12−k )−
2 ( 12−k )
1+δ 1−π2+ 1
4 ( 2( 12−k )
1+δ1−π 2+2−
1+δ1−π2
2( 12−k ) )
2
,which can be simplified by
32
writing Z≡
1+δ1−π2
2( 12−k )
=1+δ1−π2+2k
2( 12−k )
>1. Thus, expected profit is
EΠ C=π2+2( 12−k )− 1
Z+ 1
4 ( 1Z+2−Z)
2
. The total derivative with respect to 2 can be thought
of as having a direct and indirect part:
dE ΠC
dπ 2=1+ ∂ EΠC
∂Z ( −11−2 k ) . The partial derivative with
respect to Z is
∂EΠC
∂Z=Z4−2 Z3−1
2Z3, which can be factored as
∂EΠ C
∂Z=(Z−2 .11)(Z+ .717 )((Z−. 305)2+. 569 )
2 Z3. The requirements that δ 1−π2+2k≥0 and
that P1Q≤1 (or ≥-(1-3k)+3/2 1 means that pride and disappointment must be in the shaded
region in Figure 3. We also assume that e2C>0 when there are no emotions, which implies that
k< 12+√2
≃0 .2929. For the purpose of the proof we also consider values where 1=0 and 2 is
below zero but above –(2/3-2k).
33
Figure 3
Suppose that 1=0 and 2=-(2/3-2k). This implies that Z=1+ 2/3
2 ( 12−k ) . Given the factorization
of the total derivative given above, if Z<2.11, then the derivative is greater than 1. But this
inequality is equivalent to k< 1
2− 1
3×2.11≃0 .34 .
Because we assume that k cannot exceed
0.2929, this implies that
dE ΠC
dπ 2>1 .
The second derivative of the profit function with respect to
2 is
d2 EΠ C
dπ22=∂
2 EΠC
∂Z2 ( 11−2 k )
2
. The second partial derivative with respect to Z is
∂2 EΠ C
∂Z2 =12 (1+ 3
Z4 )>0 . This implies that the derivative
dE ΠC
dπ 2 starts above 1 and gets larger
as 2 increases in the interval [-(2/3-2k), 2k].
34
0
2
1
32
1
-(1-3k)
-(2/3-2k)2/3-2k 2(1-k)
2k
Suppose that we begin at a disappointment and pride (0,20) and then increase each value
by an equal amount to that we reach a new value (,20+). The value of Z does not change, but
2 becomes progressively larger. Each point in the shaded region could be increased by a small
amount holding 1 constant. That is, we move from (1,2) to (1,2+d2). This could be thought
of as decomposed into (1,2) = (,20) +(,) and (1,2+d2) = (,2
0+d)+(,). The upward
change improves profits and the equal displacement in emotional intensity leaves this
improvement unaffected. Hence, when the only change is an increase in pride in the weaker
territory 2, the profits must increase, as was to be shown.
Proof that EC decreases with 1. As above the total derivative with respect to 1 is
dE ΠC
dδ1=∂ EΠC
∂Z ( 11−2 k ) . As above,
∂EΠ C
∂Z=(Z−2 . 11)(Z+ .717 )((Z−. 305)2+. 569 )
2 Z3. Let us
limit ourselves to 2=0 and start at 1=0, where Z=1/(1-2k). The term Z-2.11 is negative if
k<(2.11-1)/2×2.11=.26. Suppose that you go to 1=2/3-2k. Then Z-2.11 is negative if k<1/2 –
1/(3×(2.11-1))=.20. Since Z is linear in 1, if k<.20,
dE ΠC
dδ1<0
for all 1[0,2/3-2k]. By
expanding each such point by (,), this shows that for k<0, the negative effect occurs in the
cross hatched region in Figure 4. To get the remainder of the region, we would have to start with
negative 1. Let 1=-2k, so that Z=1. Then clearly Z-2.11<0. Again because Z is linear in 1, this
implies that for k<.20,
dE ΠC
dδ1<0
for all 1[-2k,2/3-2k]. Hence, for k<.20, this implies that
profit decreases in disappoint for all legitimate emotion levels.
35
Figure 4
Proof of Proposition 3. Start with 1=0, so profits are EΠ Q=π2+ (1−k )2 and
EΠ C=π2+2( 12−k )− 1
Z+ 1
4 ( 1Z+2−Z)
2
where Z≡
1−π2
2 ( 12−k )
=1+2 k−π2
2 ( 12−k )
>1.
When 2=2k, Z=1 and EΠ Q=1+k2 > EΠ C=1 . At 2=0, Z=1/(1-2k) and
EΠ Q=(1−k )2 > EΠ C=(1−k )2− 8k
(1−2 k )2((1−k )2− 1
2).
36
-2k 0
2
1
32
1
-(1-3k)
-(2/3-2k)2/3-2k 2(1-k)
2k
Figure 5
At 2=-(2/3-2k), this implies that Z=1+ 2/3
2 ( 12−k ) .
As seen above the derivative of EC is
dE ΠC
dπ 2=1+ ∂ EΠC
∂Z ( −11−2 k ) where
∂EΠ C
∂Z=Z4−2 Z3−1
2 Z3. At 2k, this gives
dE ΠC
dπ 2=1+ 1
1−2 k>1
. The second derivative is
∂2 EΠ C
∂ π22=1
2 ( 3Z4+1) 1
(1−2 k )2>0
. Hence, the profit curve lies below the chord connecting the
profits at 0 and 2k, and hence EQEC for all reasonable 2 when 1=0 as seen in Figure 5
37
1
0
1
(1-k)2-8k((1-k)2-1/2)/(1-2k)2
EC
EQ
1+k2
(1-k)2
2k
E
2
When 1=0, the gap between EQ and EC is maximized when 2 is the value where
dE ΠC
dπ 2=1+∂ EΠC
∂Z ( −11−2 k )=dE ΠQ
dπ2=1
. This occurs when Z=2.11. At 2=-2/3+2k, this
requires k<0.20. Hence if this is true of k and 1=0, as 2 increases the gap between contest and
quota shrinks. Now free 1 to be positive. Any location can be reached by starting at (0,2) and
adding (,) to it. As we have seen this causes the gap to widen, but if you fix 1 and increase 2
the effect on the gap is driven by the effect at (0,2). For any 1, increase in 2 shrinks the gap.
Contest with Handicapping:
Because of the known advantage of territory 1 over territory 2, it seems unfair for the contest to
compare sales that are in part beyond the control of the salespeople. Suppose instead that the
weaker territory 2 is given an accounting head start in an amount h. That is, for the purposes of
the contest, sales are treated as though they were are s1-h and s2+h. Let the handicapped
imbalance be denoted Kk-h. The equilibrium effort levels that result are
e1¿=e2¿=12−K− K /2
12−K . The maximum efforts are achieved at K=0. By deducting this implied
handicap h=k from territory 1 and adding it to territory 2, the problem reduces to the balanced
case. For the handicapped contest the efforts are, e
1HC∗¿=e2HC∗¿
=1
¿¿
, and the profit isEπ HC∗¿=1 ¿.
Quota with Handicapping (Dual Quota Plans):
Consider territory-specific quotas. The quota territory 1, Q1, is greater than the quota for territory
2, Q2, due to the territory imbalance. The sales of territory 1 are s1=k+e1+ε1 and the sales of
territory 2 are s2=−k+e2+ε2 . The probabilities of making quotas respectively
Pr (1 makes Q1 )=12+k−Q1+e1 ,
Pr (2 makes Q2 )=
12−k−Q2+e2 .
The efforts in terms of
38
bonus are e1=e2=B . Binding at both participation constraints, quotas in terms of bonus are
Q1=12+k+ B
2 and Q2=
12−k+B
2 . So we rewrite the expected profit in terms of B only,
EΠ Q=2 B−B2 . This gives B=1 and the equilibrium, quotas, effort levels, and firm’s profit are
easily obtained. The equilibrium effort levels are e
1DQ∗¿=e2DQ∗¿
=1
¿¿
, and the firm’s profit for the
dual quota are given asEΠ DQ∗¿=1 ¿.
Thus handicapping of the contest and quota correct for territory imbalance and the
agents’ efforts and firm’s profit are those that would be obtained if territories were balanced.
Proof of Proposition 4.
Point A of Figure 1 is where δ1=−(1−3 k )+ 32 δ1 , or δ 1=2(1−3k )=π2 At point A, the contest
profits are
EΠ C=2(1−3k )+(1−k− k2( 1
2−k ))2
.
This exceeds the quota profits at 0, (1-k)2 if and
only if −(k−.25 )(k−.773 )(k−.369 )
(1−2k )2≥0
. Since we are assuming that k≤0.2<.25, this will be
true.
Proof of Proposition 5.
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Suppose that k≤0.2 and that intensity of emotions are at point A where C=2(1-3k). This
satisfies μC≥1−(1−k− k
1−2 k )2
if 2(1−3 k )≥−(1−k− k
1−2k )2
. The roots of this inequality
are 0.202 and above, so for all assumed k, this inequality holds and handicapping the contest
reduces the expected profits. In the context of quotas, for any value of k all one needs to start
with emotions near zero and handicapping will help balance the territories with only a small
reduction in emotional motivation.
40